$(a + b + c)^3 - a^3 - b^3 - c^3 = 3(a + b)(b + c)(a + c)$
$(a + b + c)^3 - a^3 - b^3 - c^3 = ((a + b + c)^3 - a^3) - (b^3 + c^3) = (a + b + c - a)((a + b + c)^2 + a(a + b + c) + a^2) - (b + c)(b^2 - bc + c^2) = (b + c)(a^2 + b^2 + c^2 + 2ab + 2ac + 2bc + a^2 + ab + ac + a^2 - b^2 + bc - c^2) = (b + c)(3a^2 + 3ab + 3ac + 3bc) = 3(b + c)(a^2 + ab + ac + bc) = 3(b + c)((a^2 + ab) + (ac + bc)) = 3(b + c)(a(a + b) + c(a + b)) = 3(b + c)(a + b)(a + c) = 3(a + b)(b + c)(a + c)$

$(a - b)^3 + (b - c)^3 - (a - c)^3 = -3(a - b)(b - c)(a - c)$
$(a - b)^3 + (b - c)^3 - (a - c)^3 = ((a - b)^3 + (b - c)^3) - (a - c)^3 = (a - b + b - c)((a - b)^2 - (a - b)(b - c) + (b - c)^2) - (a - c)^3 = (a - c)(a^2 - 2ab + b^2 - (ab - b^2 - ac + bc) + b^2 - 2bc + c^2) - (a - c)^3 = (a - c)(a^2 - 2ab + b^2 - ab + b^2 + ac - bc + b^2 - 2bc + c^2) - (a - c)^3 = (a - c)(a^2 - 3ab + 3b^2 - 3bc + ac + c^2) - (a - c)^3 = (a - c)(a^2 - 3ab + 3b^2 - 3bc + ac + c^2 - (a - c)^2) = (a - c)(a^2 - 3ab + 3b^2 - 3bc + ac + c^2 - (a^2 - 2ac + c^2)) = (a - c)(a^2 - 3ab + 3b^2 - 3bc + ac + c^2 - a^2 + 2ac - c^2) = (a - c)(-3ab + 3b^2 - 3bc + 3ac) = -3(a - c)(ab - b^2 + bc - ac) = -3(a - c)((ab - b^2) + (bc - ac)) = -3(a - c)(b(a - b) + c(b - a)) = -3(a - c)(b(a - b) - c(a - b)) = -3(a - c)(a - b)(b - c) = -3(a - b)(b - c)(a - c)$